Exercise

Miss Mary gave a polynomial $f\left(x\right)=x^3+2x^2+3x+2$ to the class and said:

"When $f\left(x\right)$ is divided by another polynomial $g\left(x\right)$, it leaves quotient $q\left(x\right)$ and remainder $r\left(x\right)$, where $q\left(x\right)=r\left(x\right)\ne 1$. All the polynomials $g\left(x\right), q\left(x\right), r\left(x\right)$ have integral coefficients."

Using the above data, find the value of $g\left(5\right)$.

Two friends Rahul and Piyush went to the roof of their house in the evening to fly kites. The moment they reached the roof they noticed four monkeys sitting there. Rahul observed that the monkeys were sitting in such a way that all four of them formed a rectangle such that the length of the rectangle was twice the breadth.

The coordinates of two of the monkeys are shown in the figure below:

Where might any one of the other two monkeys be seated, if they are seated to the left of the monkeys shown in the figure?

Shown below is the seating arrangement in a classroom in $X-Y$ plane. The teacher’s position is the origin of $X-Y$ plane.

Pooja’s position is given as $\left(1, -1\right)$, then Ravi and Surya's position (respectively) are

If distance of two points is equal from the $Y$-axis and the line joining them is parallel to $X$-axis, then what is the sum of their abscissa?

Constructed externally on the sides $AB, AC$ of $△ABC$ are equilateral triangle $ABX$ and $ACY$. If $P, Q, R$ are the mid-ponts of $AX, AY$ and $BC$ respectively, then $△PQR$ is

In the parallelogram $ABCD$ of the given figure, $\angle PAQ$ is an obtuse angle. Two equilateral triangles $ABP$ and $ADQ$ are drawn outside the parallelogram. Prove that $△CPQ$ is also an equilateral triangle.

A road is passing parallel to the diagonal a square field $ABCD$ as shown in the figure below. Two cars are moving in parallel along $AE$ and $CF.$ If the measure of $\angle BAE$ is $20°$ then what is the measure of $\angle BCF?$

The picture below is a component of structures used for mast, towers. The smallest element in these designs is a triangle, which increases stability and strength to the structures.

Discuss why triangles used as the elementary shapes instead of quadrilaterals such as square or parallelograms or any other polygons in the design.